Taking into account another central ingredient that is present
in ecological systems, namely, the competition with other
individuals in the neighborhood for resources, the formation of
periodic spatial structures was observed in
Refs. [1]}, [2]}, [3]}. In these
nonlocally interacting Brownian bug models it was assumed
that the reproduction probability depends on the number of
other organisms in the neighborhood. In
Ref. [4]} nonlocally interacting Lévy
bugs, i.e., reproducing and dying organisms that undergo
Lévy flights, were studied. This type of motion is relevant
to model cell migration [5]}, biological
searching strategies [6]}, [7]}, bacteria
dynamics [8]}, or pattern formation of
mussels [9]}. In Ref. [4]} it was
shown that the formation of a periodic pattern is robust with
respect to the type of spatial motion that the particles
perform. The periodic arrangement of clusters in these
nonlocally interacting bug models is a consequence of the
competitive interaction and has a spatial scale determined by
the interaction range [1]}. However, a deeper
analysis of the differences and similarities between the
Brownian and Lévy cases is still missing. In particular, as
shown in [2]}, [13]}, this analysis can be
very conveniently performed by considering the limit of the
interaction distance reaching the system size (global
interaction), since a unique cluster appears which helps to
understand and characterize the cluster properties, and the
fluctuations of the population size.
